Advanced option for forcing 6-fold symmetry (hexapole, sextupole)

CPO3D does not have the reflection planes required for 6-fold symmetry.

Therefore to simulate a system that has 6-fold symmetry the electrodes and their subdivisions have to be carefully chosen to maintain this symmetry.

The transverse fields near the axis of such a system can be very weak. Because these fields involve the near-cancellation of the larger fields of the individual electrodes the errors in the near-axis fields can be larger than those expected for the requested inaccuracy level. So even when the electrodes and their subdivisions have strict 6-fold symmetry the fields near the axis might not have this symmetry exactly. This can be important when the aberrations of lenses are being studied.

Therefore an advanced option has been added to force the fields (and potentials) to have exact 6-fold symmetry. This option can be activated from the ‘symmetries’ page.

The 6-fold symmetry can be symmetric (++++++) or anti-symmetric (+-+-+-) around the z axis.

Examples are given in shap3d29, shap3d14 and xmpl3d93.

Technical note:

In this option the program rotates the point at which a field is required to a point in the ‘primary sector’, which is the sector that has an angular range from -30 to +30 degrees with respect to the y axis. It is assumed that the axis of the system is the z axis and that the xz plane (that is, the y = 0 plane) is a plane of voltage symmetry. The program calculates the field (or potential) at this point and then rotates the field back to the original direction (taking account of any changes in sign of the fields and potentials).

Back to general note on symmetries.